The angle of elevation of top of the tower from a point P on the ground is `30^(@)` . If the points is 45 m away from the foot of the tower , then the height of the tower is

Options

A. 45 m
B. 15 m
C. `15 sqrt(3)`
D. `20 sqrt(3) m`

Answer

To find the height of the tower based on the given information, we can follow these steps:

### Step 1: Understand the problem
We have a point P on the ground from which the angle of elevation to the top of the tower is given as \(30^\circ\). The distance from point P to the foot of the tower is 45 meters.

### Step 2: Draw a diagram
Visualize the scenario by drawing a right triangle where:
- The height of the tower is the perpendicular side (let's call it \(h\)).
- The distance from point P to the foot of the tower is the base (45 m).
- The angle of elevation from point P to the top of the tower is \(30^\circ\).

### Step 3: Set up the trigonometric relationship
In a right triangle, the tangent of an angle is defined as the ratio of the opposite side (height of the tower) to the adjacent side (distance from point P to the foot of the tower). Therefore, we can write:

\[ \tan(30^\circ) = \frac{h}{45} \]

### Step 4: Substitute the value of \(\tan(30^\circ)\)
We know that:

\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

Substituting this into the equation gives:

\[ \frac{1}{\sqrt{3}} = \frac{h}{45} \]

### Step 5: Solve for \(h\)
To find \(h\), we can rearrange the equation:

\[ h = 45 \cdot \frac{1}{\sqrt{3}} = \frac{45}{\sqrt{3}} \]

### Step 6: Rationalize the denominator
To express \(h\) in a more standard form, we can rationalize the denominator:

\[ h = \frac{45}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{45\sqrt{3}}{3} \]

### Step 7: Simplify
Now, simplify the fraction:

\[ h = 15\sqrt{3} \]

### Final Answer
Thus, the height of the tower is:

\[ h = 15\sqrt{3} \text{ meters} \]

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